ON A PROBLEM CONNECTED WITH BETA AND GAMMA DISTRIBUTIONS BY R. G. LAHA(i)
1. Introduction. The random variable X is said to have a Gamma distribution G(x;0,a)if
du for x > 0, (1.1) P(X = x) = G(x;0,a) = JoT(a)" 0 for x ^ 0, where 0 > 0, a > 0. Let X and Y be two independently and identically distributed random variables each having a Gamma distribution of the form (1.1). Then it is well known [1, pp. 243-244], that the random variable W = X¡iX + Y) has a Beta distribution Biw ; a, a) given by 0 for w = 0,
(1.2) PiW^w) = Biw;x,x)=\ ) u"-1il-u)'-1du for0
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 288 R. G. LAHA (November
2. Some general properties of distribution laws. We first prove a lemma which is instrumental in the proof of the subsequent theorem.
Lemma 2.1. Let X and Y be two independently and identically distributed random variables with a common distribution function Fix) which is continuous at the point x = 0. Suppose that the quotient V = XjY has a one-sided distribution F0iv) such that F0(u) = 0for v^O. Then X has also a one-sided distribution.
Proof. As usual we assume that each of the distribution functions F(x) and F0iv) is everywhere continuous to the right so that for every v > 0, we have
Foi0) - Foi - v) = Prob( - v< XjY < 0) (2.1) = JJ [F(0)-F(-!;y)]dF(y)
+ f [F(-t;y-0)-F(0)]dF(y). J - 00
From the conditions of the lemma we have Fo(0) = F0( — i;) = 0 for every v > 0 so that (2.1) gives
fiOO (2.2a) [F(0)-F(-i;y)]dF(y) = 0, Jo
(2.2b) f [F( -vy-0)- F(0)]dFiy) = 0 J-00 holding for all v > 0. Then it follows immediately from (2.2a) that either Fiy) = 0 for all y = 0 or Fiy) = 1 for all y ^ 0. It is also easy to verify that the relation (2.2b) is satisfied in either of the above cases.This completes the proof of the lemma. We next prove the following theorem:
Theorem 2.1. Let X and Y be two independently and identically distributed random variables with a common distribution function Fix). Let the random variable W = XfiX + Y) follow the Beta distribution of the form (1.2). Then Fix) has the following general properties: (1) Either Fix) = Ofor x^Oor Fix) = 1for x ^ 0, (2) F(x) is absolutely continuous and has a continuous probability density function p(x) = F\x) > 0.
Proof. It is easy to verify from the conditions of the theorem that the quotient V = XjY = W/il - W) has the distribution function F0(t>)given by
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1964] BETA AND GAMMA DISTRIBUTIONS 289 r r(2a) ua_1 -du fory>0, (2.3) F0iv) = r(a)r(a) (1 +«)2« 0 for v < 0. Thus property ( 1) follows immediately from Lemma 2.1. Therefore, we can assume without any loss of generality that F(x) = 0 for x = 0, that is, the random variable X is positive. After some elementary integration we can easily derive from (2.3) the characteristic function of the distribution of In F as:
Ft ¿tin v, _ Tix + it)rjx-it) He ] - Tix)Tix) •
We denote the characteristic function of the distribution of In ATby ^(f) and thus obtain the basic equation T(a + ¿r)r(a- it) (2.4)
Lemma 3.1. Let /(z) be an analytic characteristic function which is regular in a certain horizontal strip of maximum width containing the real axis. Then the purely imaginary points on the boundary of this strip of regularity are singular points offz). This lemma is due to Levy [5]. A proof of this lemma is given by Lukacs in [8].
Lemma 3.2. Let /(z) be a decomposable characteristic function which is regular in a strip — ß < Imz < ß iß >0) of the complex z-plane. LetfAz) be a factor offiz). Then the characteristic function fxiz) is also regular at least in the same strip. This lemma on the factorization of analytic characteristic functions is due toRaikov[10]. Lemma 3.3. Let Riz) be an entire function of order unity having only purely imaginary zeros and R(0) = 1. Then its reciprocal 1/R(z) is always a charac- teristic function.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 290 R. G. LAHA [November The proof of this lemma is given in [6, p. 140].
Lemma 3.4. Letfz) be an analytic characteristic function and suppose that fo(z) = [/(z) +/( - z)]/2 andfiz) = [/(z) -/( - z)]/2 are respectively the even and odd parts offiz). Let p,p0 and px denote respectively the radii of convergence of Maclaurin series for /(z),/0(z) andfxiz). Then p = p0 = px. This lemma has been proved by the author in [3].
Lemma 3.5. Let fz) be the characteristic function of an infinitely divisible distribution which is regular in a certain horizontal strip. Then /(z) has no zeros inside its strip of regularity and therefore (z)= ln/(z) is defined and regular in the same strip. Moreover, the function 0(z) = Lemma 3.6. Let p(x) be a continuous non-negative function of the real variable x. Let the integral J^ x"p(x) dx (y reaO exist for all v in the interval 0 < v < V iV> 0).Then the integral Iiz)= f0°°x~lzp(x)dx as a function of the complex variable z is regular in the strip 0 < Imz < Vof the upper half-plane. This lemma has been proved by the author in [4]. As a special case of this lemma, we note that if the integral J" x"pix)dx exists for all real v > 0, then the integral I{z) is regular throughout the upper half-plane Imz > 0. Lemma 3.7. Let the distribution function F(x) of a positive random variable X be absolutely continuous and have a continuous probability density function p(x) = F'(x) > 0. Let Fix) possess finite absolute moments up to a certain order ô (á > 0 not necessarily an integer). Let rpit) = Eie"inX) denote the characteristic function of the distribution oflnX. Then the function ¡¡/i-z) = Eie-izlnX) considered as a function of the complex variable z is regular in the strip 0 < Imz < 5 + £ (s > 0 may be a suficiently small number). Conversely if the function \pi — z) is regular in the strip 0 iK - z) = £(e"izInX ) = r x~izpix)dx. Then the proof follows at once from Lemma 3.6. The proof of the converse statement is immediate, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1964] BETA AND GAMMA DISTRIBUTIONS 291 As a special case of Lemma 3.7, it follows that if F(x) has finite moments of all orders, then the function i/r( - z) is regular throughout the upper half-plane lmz>0. 4. A characterization of a class of distribution laws. In this section, we derive a characterization of a class of distribution laws having finite moments up to a certain order. Then we shall obtain a characterization of the Gamma distribution as a particular case. First, we prove the following theorem : Theorem 4.1. Under the same conditions as in Theorem 2.1, the distribution function Fix) of the random variable X always has finite absolute moments of order y where 0 < y z = iyr where yr = a + r, r = 0,1,2,•••, all located on the imaginary axis in the upper half-plane. Therefore, we canuse the factorization theorem of Hadamard [11, p. 250] and obtain (4.1) rrT(a\ ^ = e"» nil - -£-)*""> Tix + iz) rl=o\ i?, / Here it is easy to verify that p ^ 0 is a real constant. Similarly the function 1/T(a —iz) is an entire function of order unity having only purely imaginary zeros at the points: z = — ¿yr where yr = x + r, r = 0,1,2,"-. Therefore it has a canonical representation similar to (4.1). Thus we have for complex values of z r4r. r(a + iz)r(a-iz) » _1_ ^'; r(«)r(a) rUd+^/jf) where yr = a + r (a > 0, r = 0, 1,2, •••). Then we conclude at once from the relation (4.2) that the characteristic function on the right-hand side of (2.4) can be continued in the complex z-plane and this function as a function of the complex variable z is regular in the strip | Im z | < a. Then we apply Lemma 3.2 on the factorization of analytic characteristic functions to the relation (2.4) and deduce at once that the function ij/it) can also be con- License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 292 R. G. LAHA [November tinued in the complex z-plane and the function \¡>iz)is regular at least in the strip |Imz| < a. Thus \j/i — z) is also regular in the strip | Imz| < a and the proof follows at once from Lemma 3.7. Theorem 4.2. Under the same conditions as in Theorem 2.1,/ef the distribution function Fix) have finite absolute moments up to order a + ô (<5^ 0). Let ¡bit) denote the characteristic function of the distribution o/ln X, then the function tfri — z), as a function of the complex variable z, is regular in the strip — ot The proof of this theorem follows immediately when we combine the results of Lemma 3.7 and Theorem 4.1. As a special case of this theorem, we note that if the distribution function F(x) has finite moments of all orders, then the function \j/i ?*•z) is regular throughout the region Im z > - a ( a > 0) of the complex z-plane. First we discuss a method of constructing a class of infinitely many distri- bution functions Fix) having finite absolute moments up to a certain order a + <5(<5jä0), where the random variable W = X/iX + Y) has a Beta distribution of the form (1.2) For this purpose we examine more closely the basic equation where yr= a + r (oc>0; r = 0,1,2, •••), assuming that (4.3) holds for all complex values of z in the strip | Im z | < a. It is also easy to verify that the condition Er=ol/y2 < oo is satisfied. We divide the entire set of non-negative integers (0,1,2, •••) into two mutually exclusive and exhaustive subsets and denote by ir) = irx,r2,r3,--) an arbitrary subset of non-negative integers, finite or denumerable, and by is) =(j,,S2,Sj,-") the corresponding complementary subset such that (r)+(s) = (0,1,2,-). As a special case, any one of these subsets (r) and (s) may constitute the whole set of non-negative integers (0,1,2, •••). In such a case the other subset reduces to an empty set. With the sets (r) and (s) so defined, we construct the entire functions of order unity as follows : License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1964] BETA AND GAMMA DISTRIBUTIONS 293 (4.4) where yrj = x + r¡ and ySj = x + Sj. It is then easy to verify that (4.5) Piz)Qiz)Pi - z)Qi- z) = ft (l + 3) r = 0 \ 7r / We now write (4.6) ipiz) P(z)S(-z) where P(z) and g(z) are already defined in (4.4) and ß is an arbitrary real number. We verify immediately from Lemma 3.3 that i/r(z) is a characteristic function. We also note that in view of the relation (4.5) the characteristic function i^(z) which is regular in a certain horizontal strip containing the real axis satisfies the basic equation (4.3). Thus the function \piz) yields a distribution function F(x) with the desired property. We next discuss the arrangement of elements in the sets (r) and (s) when it is given that the distribution function F(x) has finite absolute moments up to order x + ö ((5 _ 0). Without any loss of generality, we can assume that the integer elements in the sets (r) and is) are arranged in increasing order of magnitude. It is easy to verify after some elementary arguments that in this case the arrange- ment of the elements in the sets (r) and (5) is as follows : (47) (r) = (0,1,2,--IS]- ), is) = (M + l,...), that is, the first [<5]+ 1 elements of the set (r) are 0,1,2,-•• [ó] and the first element of the set (s) is [<5] +1, while the arrangement of the remaining integers in the sets (r) and (s) is left entirely arbitrary. Here [<5]denotes the largest integer con- tained in ö. We can now summarize the main result in the form of the following theorem : Theorem 4.3. Let X be a positive random variable such that the charac- teristic function \¡>iz)of the distribution of\nX is given by (4.6) where P{z) and Qfz) are defined as in (4.4). Let the integer elements in both sets (r) and (s) be arranged in increasing order of magnitude and further satisfy' the condition in (4.7). Any such construction ofipiz) yields a distribution function F(x) having finite absolute moments up to order a + ö (á _ 0) where the random variable XjiX + Y) has a Beta distribution of the form (1.2). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 294 R. G. LAHA [November From now on we denote this class of distribution functions by Cx+S. We are now in a position to prove the following theorem on a characterization of the class Ca+S. Theorem 4.4. Let the following two conditions be satisfied in addition to the restrictions of Theorem 2.1: (1) The distribution function Fix) of the random variable X has finite absolute moments up to order a + ö (5 _ 0). (2) The random variable ln.Y has an infinitely divisible distribution. Then Fix) belongs to the class Ca+i. Proof. We write i/r0(z) = 1/P(z) Qi~ z) where Piz) and g(z) are defined in (4.4). First we note that the characteristic function i/f0(z) is infinitely divisible, as it is obtained as the limit of a sequence of infinitely divisible characteristic functions. Let the integer elements in the sets (r) and (s) corresponding to i/s0iz) be arranged in increasing order of magnitude and further satisfy the condition (4.7). Then ^0( — z) is regular in the strip —a < Im z < a + [<5]+ 1 and has no zeros inside this strip. It is also easy to verify from the conditions of the theorem that the characteristic function ip(— z) = E(e~lzinX) is also regular in the strip — a < Imz < a + [ó"] + 1 and has no zeros inside this strip. We write (4.8) 1A(z)= and verify easily that the function 0( — z) is also regular in the strip — (4.9) 0(z)0(-z) = l. From the relation (4.9) we can verify easily that the function InO(z) is also regular in the strip | Im z | < a + [<5] + 1. We further conclude from (4.9) that In 0(z) is an odd function, that is, (4.10) ln0(-z)=-lnö(z). Next we multiply both sides of (4.8) by ^0( —z) and obtain (4.11) where (biz) = ^(z)i/r0( - z) and <£0(z) = i¡/0(z)\¡/0( - z). Here we note that each of the functions License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1964] BETA AND GAMMA DISTRIBUTIONS 295 /(z)=i.¿,n^(z)) 1 d2 /o(z) = T'dz^ln^o(z)' /!<«)-7-eta«'). where In cbiz) dz2 z = 0 and deduce easily from the equation (4.11) the relation (4.12) /(z) = /0(z) +fAz). We now use Lemma 3.5 and verify easily from (4.12) that the functions/0(z) and fAz) are respectively the even and odd parts of the characteristic function/(z). Here/0(z) is only regular in the strip |lmz| < a, while fAz) is regular in the larger strip | Imz| < a + [<3]+ 1. But according to Lemma 3.4,thisisnot possible. Therefore /(z) = 0 for all z so thatln 0(z) = ißz. We further note that each of the functions cbiz) and where ß is an arbitrary real number. Thus we obtain finally from (4.8) the equation (4.13) ifrz) = Piz)Qi-z) where Piz) and Q(z) are defined in (4.4). Then the proof of the theorem follows immediately, since the integer elements in the sets (r) and (s) corresponding to Piz) and Qiz) in (4.13) satisfy the condition in (4.7). The following characterization of the Gamma distribution is an immediate consequence of Theorem 4.4. Theorem 4.5. Let the following conditions be satisfied in addition to the restrictions of Theorem 2.1 : (1) The distribution function Fix) of the random variable X has finite absolute moments of all orders. (2) The random variable ln X has an infinitely divisible distribution. Then Fix) is a Gamma distribution of the form (1.1). Proof. It follows at once from Theorem 4.4 that under the conditions of the theorem the set (r) constitutes the whole set of non-negative integers (0,1,2,--) while the set (s) reduces to an empty set. As a consequence we obtain from (4.1) the relation License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 296 R. G. LAHA [November (4.14) ^(z)=r(r(a)2) holding for all complex z in the region Im z < a, where y is an arbitrary real number. We next substitute z = — iv (t> > 0 real) in (4.14) and obtain (4.15) H - iv) = f° x vpix)dx = ^V? e>\ Jo l W Finally we set v = k (fc > 0 an integer) and obtain (4.16) pk = £Vp(x)dx = r(^}fe) 1 (0 = e~>> 0). Here pk is the moment of order k of the distribution function Fix). The proof of Theorem 4.5 follows immediately from the fact that the moments in (4.16) uniquely determine the Gamma distribution G(x; 9, a) given in (1.1). 5. Further examples. In this section, we shall construct another class of infinitely many characteristic functions i/^z) which satisfy the basic equation (4.3) and for which the corresponding distribution functions Fix) have finite absolute moments up to order a + 5 (<5^0). But we shall show that the charac- teristic functions \¡>iz)so constructed are not infinitely divisible and consequently we shall verify that the condition (2) of Theorem 4.4 as well as of Theorem 4.5 is essential. For this purpose we require the following lemma: Lemma 5.1. Let K)('*r) (5.1) f(t) (-tK'-sX'-t) where p = y + iö, p = y — iô and y > 0; ô > 0 are both real. Then fit) is always a characteristic function whenever the relation ô ^ 2 yj2y is satisfied. The proof follows from a more general result on rational characteristic functions given by Lukacs in [7]. We next define the quantities: -yrj = a + rj] j = 1,2,---, Nx; Nx + 1,—, (5.2a) ^ = 2V2yrj; ;. l,2,-,Nl5 Hrj = yrj + itrj', j = l,2,-,Nx, .ßrj = "¡r.-tö,,; i = i,2,-,Nx, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1964] BETA AND GAMMA DISTRIBUTIONS 297 and ySJ = x+Sj-, j=l,2,-,N2;N2 + l,—, ÔS]= 2V2ySj; j = l,2,-,N2, (5.2b) Psj = Vsj + iosj-, j = l,2,-,N2, IPsj = yS]~ioSj; j = l,2,-,N2, corresponding to the elements of the sets (r) and (,s)where the positive integers Nx and jV2 may be either finite or infinite. We verify easily that the condition Z"o 1/|Pr\2 < LT-ol/y,2 < °° is satisfied. We now construct the function \piz) as (5.3) |l-i)(l-i), \ »ft, / \ »fey/ X 11 / t\/ t\/ t\ 11 7 = 1 If both the numbers Nx and iV2are finite we conclude immediately from Lemmas 3.3 and 5.1 that the function i/^z) in (5.3)is a characteristic function. On the other hand, if either of the numbers Nx or JV2is infinite, we can also verify easily that i/f(z)is a characteristic function, using the continuity theorem of Levy. We further note that the characteristic function i^(z) in (5.3) satisfies the basic equation (4.3) and the function t/f( — z) is regular in the strip — yn < Imz < ySi. Thus in this case we can again obtain a theorem which is analogous to Theorem 4.3.Therefore we have constructed another class of infinitely many distribution functions F(x) having finite moments up to a certain order and possessing the desired property. But we note at the same time that the characteristic function License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 298 R. G. LAHA References 1. H. Cramer, Mathematical methods of statistics, Princeton Univ. Press, Princeton, N.J. 1946. 2. F. M. C. 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Nauk SSSR Ser. Mat. 2 (1938), 91-124. (Russian) 11. E. C. Titchmarsh, The theory of functions, Oxford Univ. Press, Oxford, 1939. The Catholic University of America, Washington, D.C License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use